No abstract

Based on the scalar–tensor–vector modified gravitational theory, a modified gravity Schwarzschild black hole solution has been given in the existing literature. Such a black hole spacetime is obtained through the inclusion of a modified gravity coupling parameter, which corresponds to the modified gravitational constant and the black hole charge. In this sense, the modified gravity parameter acts as not only an enhanced gravitational effect but also a gravitational repulsive force contribution to a test particle moving around the black hole. Because the modified Schwarzschild spacetime is static spherical symmetric, it is integrable. However, the spherical symmetry and the integrability are destroyed when the black hole is immersed in an external asymptotic uniform magnetic field and the particle is charged. Although the magnetized modified Schwarzschild spacetime is nonintegrable and inseparable, it allows for the application of explicit symplectic integrators when its Hamiltonian is split into five explicitly integrable parts. Taking one of the proposed explicit symplectic integrators and the techniques of Poincaré sections and fast Lyapunov indicators as numerical tools, we show that the charged particle can have chaotic motions under some circumstances. Chaos is strengthened with an increase of the modified gravity parameter from the global phase space structures. There are similar results when the magnetic field parameter and the particle energy increase. However, an increase of the particle angular momentum weakens the strength of chaos.

The motion of photons around the Kerr-Newman-dS black hole surrounded by quintessence and a cloud of strings is investigated. The existence of the Carter constant leads to that of unstable circular photon orbits on a two-dimensional plane not limited to the equatorial plane and unstable spherical photon orbits in the three-dimensional space. These circular or spherical photon orbits can determine two impact parameters, which are used to calculate black hole shadows. For the case of a spherically symmetric nonrotating black hole, the black hole shadow is circular and its size is independent of an observation angle and a plane on which a circular photon orbit exists. The shadow sizes are significantly influenced by the parameters involving the cloud of strings, quintessence parameter, magnitude of quintessential state parameter , and cosmological constant. When the black hole is spinning and axially symmetric, the black hole shadow is dependent on the observation angle. The effects of the parameters excluding the spin parameter on the sizes of black hole shadows in the rotating case are similar to those in the nonrotating case. Based on the Event Horizon Telescope observations of M87*, the constraint of the curvature radius is used to constrain these parameters. For slowly rotating black holes, the allowed regions of the parameters including the cosmological constant are given.